ar X iv : 0 90 3 . 54 18 v 1 [ m at h - ph ] 3 1 M ar 2 00 9 Factor - Group - Generated Polar Spaces and ( Multi - ) Qudits

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group G, we first construct vector spaces over GF(p), p a prime, by factorising G over appropriate normal subgroups. Then, by expressing GF(p) in terms of the commutator subgroup of G, we construct alternating bilinear forms, which reflect whether or not two elements of G commute. Restricting to p = 2, we search for “refinements” in terms of quadratic forms, which capture the fact whether or not the order of an element of G is ≤ 2. Such factor-groupgenerated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a “condensation” of several distinct elements of G. Finally, several well-known physical examples (singleand two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism. Mathematics Subject Classification (2000): 20-xx – 51A50 – 81R05 Key-words: Groups – Symplectic and Orthogonal Polar Spaces – Geometry of Generalised Pauli Groups